Chapter 1
Electric Charges, Forces, and Fields
1
Units of Chapter 1

Electric Charge

Insulators and Conductors

Coulomb’s Law

The Electric Field

Electric Field Lines

Electric Fields Generated by simple distributions of charges

Shielding and Charging by Induction
Electric Charges, Forces, and Fields
Electric Charge
3
1-1 Electric Charge
The effects of electric charge were first observed as static electricity:
After being rubbed on a piece of fur,
an amber rod acquires a charge and
can attract small objects.
1-1 Electric Charge
Charging both amber and glass rods shows that there are two types
of electric charge; like charges repel and opposites attract.
1-1 Electric Charge

Electric charge is a characteristic property of many subatomic
particles.

The charges of free-standing particles are integer multiples of
the elementary charge e; we say that electric charge
is quantized.

All electrons have exactly the same charge; the charge on the
proton (in the atomic nucleus) has the same magnitude but the
opposite sign:
1-1 Electric Charge

In 1902, Joseph John Thomson proposed
the first physical model of the atom.
 He imagined that an atom were a
sphere of fluid matter positively
charged (protons and neutrons
had not yet been discovered) in
which electrons (negative) were
immersed making the whole
atom neutral.
1-1 Electric Charge

The
Geiger–Marsden
experiment

Top: Expected results: alpha particles
passing through the plum pudding
model of the atom with negligible
deflection.

Bottom: Observed results: a small
portion of the particles were deflected by
the concentrated positive charge of the
nucleus.
1-1 Electric Charge


Rutherford
overturned
Thomson's model in 1911
with his well-known gold
foil experiment in which he
demonstrated that the
atom has a tiny, heavy
nucleus.
The electrons in an atom
are in a cloud surrounding
the nucleus, and can be
separated from the atom
with relative ease.
1-1 Electric Charge


Rutherford's model is not in accord with the laws of classical physics: according to the
electromagnetic theory, a charge that is accelerated emits energy in the form of
electromagnetic radiation.
For this reason, the electrons of the atom of Rutherford, which move in circular
motion around the nucleus, would emit electromagnetic waves and then, losing
energy, annihilate the nucleus itself (theory of collapse), which is clearly not the case.
1-1 Electric Charge


Electrons of an atom are attracted to the protons in an atomic nucleus
by this electromagnetic force. The protons and neutrons in the nucleus
are attracted to each other by a different force, the nuclear force, which
is usually stronger than the electromagnetic force repelling the positively
charged protons from one another.
The classical physics of Newton does not explain many of the properties
and behavior of atoms and sub-atomic particles: the field of quantum
mechanics has been developed to better do so.
1-1 Electric Charge
When an amber rod is rubbed with fur,
some of the electrons on the atoms in
the fur are transferred to the amber:
1-1 Electric Charge

We find that the total electric charge of the universe is a
constant:
Electric charge is conserved

Also, electric charge is quantized in units of e.

The atom that has lost an electron is now positively
charged – it is a positive ion

The atom that has gained an electron is now negatively
charged – it is a negative ion
1-1 Electric Charge
Some materials can become
polarized – this means that
their atoms rotate in response
to an external charge. This is
how a charged object can
attract a neutral one.
1-2 Insulators and Conductors

Conductor: A material whose conduction electrons are free to move
throughout.


Insulator: A material whose electrons seldom move from atom to
atom.


Most metals are conductors.
Most insulators are non-metals.
Semiconductor: A material that has an electrical conductivity value
between a conductor and an insulator.

Semiconductors are the foundation of modern electronics.

The understanding of the properties of a semiconductor relies on quantum
physics to explain the movement of electrons and holes in a crystal lattice.
1-2 Insulators and Conductors
If a conductor carries
excess charge, the
excess is distributed
over the surface of the
conductor.
1-3 Coulomb’s Law

Coulomb’s law gives the force between two point charges:

The force is along the line connecting the charges, and is
attractive if the charges are opposite, and repulsive if the
charges are like.
1-3 Coulomb’s Law
The forces on the
two charges are
action-reaction
forces.
1-3 Coulomb’s Law
If there are multiple point charges, the forces add by
superposition.
1-3 Coulomb’s Law
Coulomb’s law is stated in terms of point charges, but it is
also valid for spherically symmetric charge distributions, as
long as the distance is measured from the center of the
sphere.
1-4 The Electric Field

What is a “Field” ?


A field is a physical quantity that has a value for each point in space and
time;
A field may be thought of as extending throughout the whole of space.




In practice, the strength of most fields has been found to diminish with
distance to the point of being undetectable.
For instance, in Newton's theory of gravity, the gravitational field strength is
inversely proportional to the square of the distance from the gravitating object.
Defining the field as "numbers in space" shouldn't detract from the idea
that it has physical reality. "It occupies space. It contains energy.
The field creates a "condition in space“ such that when we put a particle
in it, the particle "feels" a force.
1-4 The Electric Field

The map in the figure is a representation of a scalar field: a field of temperature.

At each point of the represented space is associated the value of a scalar quantity, the
temperature.
1-4 The Electric Field


In a vector field , at each point is associated a vector!
For example, in a weather forecast, the wind velocity is described by
assigning a vector to each point on a map. Each vector represents the
speed and direction of the movement of air at that point.
1-4 The Electric Field

The development of the independent concept of a field
truly began in the nineteenth century with the
development of the theory of electromagnetism.

The independent nature of the field became more
apparent with James Clerk Maxwell's discovery that
waves in these fields propagated at a finite speed.
Consequently, the forces on charges and currents no
longer just depended on the positions and velocities of
other charges and currents at the same time, but also on
their positions and velocities in the past.
1-4 The Electric Field

Definition of the electric field:

Here, q0 is a “test charge” – it serves to allow the electric force
to be measured, but is not large enough to create a significant
force on any other charges.
1-4 The Electric Field
If we know the electric field, we can calculate the force on any
charge:
The direction of the force
depends on the sign of the
charge – in the direction of
the field for a positive
charge, opposite to it for a
negative one.
1-4 The Electric Field
The electric field of a point charge points radially away
from a positive charge and towards a negative one.
1-4 The Electric Field
Just as electric forces can
be superposed, electric
fields can as well.
1-4 The Electric Field
1-4 The Electric Field


From the symmetry of Fig. 22-7c, we realize
that the equal y components of our two
vectors cancel and the equal x components
add.
Thus, the net electric field at the origin is in
the positive direction of the x axis and has
the magnitude
1-5 Electric Field Lines

Electric field lines are a convenient way of
visualizing the electric field.

Electric field lines:
1. Point in the direction of the field vector at every point
2. Start at positive charges or infinity
3. End at negative charges or infinity
4. Are more dense where the field is stronger
1-5 Electric Field Lines

The charge on the right is twice the magnitude of the charge on the
left (and opposite in sign), so there are twice as many field lines, and
they point towards the charge rather than away from it.
1-5 Electric Field Lines

Combinations of charges.

Note that, while the lines are less dense where the field is weaker, the field
is not necessarily zero where there are no lines. In fact, there is only one
point within the figures below where the field is zero – can you find it?
1-5 Electric Field Lines

A parallel-plate capacitor consists
of two conducting plates with
equal and opposite charges. Here
is the electric field.

This is an example of a uniform
electric field.
1-5 Electric Field Lines
The Electric Field due to an Electric Dipole
The Electric Field due to an Electric Dipole
p = qd

Is a vector quantity known as the electric dipole moment of
the dipole

A general form for the electric dipole field is
E k


3 rˆ ( p  rˆ )  p
r
3
The Electric Field due to a Continuous Charge:
1-6 Electric Fields Generated by simple distributions of charges

Electric Field on the Axis of
Charged Ring

The net E-field (on the axis) is
along the axis, outwards from
the ring if the charge Q is
positive and towards the ring
in the charge Q is negative.

The charge density  on the
ring is just the total charge Q
on the ring divided by its
circumference 2/R.
1-6 Electric Fields Generated by simple distributions of charges

Treating the differential charge dq as a point charge the differential
electric field at a distance z from the center of the ring (the origin) is given by
1-6 Electric Fields Generated by simple distributions of charges

The z component of the Electric Field is

Since the sum of the y-components cancel out, the magnitude of the electric field is
equal to the sum of the z-components of the E-field. We can express this as an
integral over the differential arc length ds
1-6 Electric Fields Generated by simple distributions of charges

Limit case – At very large distance compared to the ring radius the field is…

… the same as that of a point charge, as we expected!
Example, Electric Field of a Charged Circular Rod
Example, Electric Field of a Charged Circular Rod
Example, Electric Field of a Charged Circular Rod
Our element has a symmetrically located
(mirror image) element ds in the bottom
half of the rod.
If we resolve the electric field vectors of ds
and ds’ into x and y components as shown
in we see that their y components cancel
(because they have equal magnitudes and
are in opposite directions).We also see that
their x components have equal magnitudes
and are in the same direction.
Example, Electric Field of a Charged Circular Rod
The Electric Field due to a Charged Disk

Let’s consider a small ring

From the previous example we easily find
in this case the integration variable is r
The Electric Field due to a Charged Disk
The Electric Field due to a Charged Disk

If we let R →∞, while keeping z finite, the
second term in the parentheses in the
above equation approaches zero, and
this equation reduces to
The Electric Field due to a Charged Disk

If we let R/z →0, we have (Taylor approximation)…
Motion of a charged particle in a uniform electric field



The motion of a charged particle in an electric field is in general very
complex;
The problem is simple if the electric field is uniform , such as that
which is located between two flat plates that carry equal and
opposite charges;
If a particle of charge q and mass m starts with initial speed 0, or
has an initial velocity parallel to the lines of the electric field , its
motion is similar to that of a body subjected to the weight force,
because it acts on a constant acceleration, that is, for the second
law of dynamics…
a 
F
m

qE
m
Motion of a charged particle in a uniform electric field

An important case is that of
located in a uniform electric
moves from the starting point
line which is A .Let’s find the
particle

Denoting by W the work done by the electric force acting on the
particle, the energy theorem states that for the particle we have
1
2
mv
2
a particle of charge q and mass m that is
field with zero initial speed. In this case it
A to point B the final place on the same field
final velocity v acquired in this way from the
 W A  B  q V A  V B   v 
2 q V A  V B
m

Motion of a charged particle in a uniform electric field


Consider a particle with charge q and mass m that enters between two
plates charged with opposite signs, with the velocity vector v0 parallel to
the plates. To fix ideas, in the figure the armatures are horizontal and the
particle moves to the right.
Once that is located between the plates, a force acts on the charge F =
qE pointing towards the top of the figure, and then perpendicular to v0.
For the second law of motion we can write
•
a 
F
m

qE
m
Motion of a charged particle in a uniform electric field



The particle is subjected to two simultaneous motions; in fact, the particle
moves:
According to the law of uniform motion in the direction of v0 (for the
principle of inertia, since there are no forces parallel to v0);
Uniformly accelerated motion along the direction of a.
This is a situation physically identical to that of a stone thrown
horizontally near the ground subject to gravity alone
 x (t )  v 0t

1 qE 2

y (t ) 
t

2 m
A Dipole in an Electric Field

Although the net force on the
dipole from the field is zero, and
the center of mass of the dipole
does not move, the forces on the
charged ends do produce a net
torque t on the dipole about its
center of mass.
A Dipole in an Electric Field


The center of mass lies on the
line connecting the charged ends,
at some distance x from one end
and a distance d -x from the other
end.
The net torque is:
A Dipole in an Electric Field
Potential Energy

Potential energy can be associated
with the orientation of an electric
dipole in an electric field.

The dipole has its least potential
energy when it is in its equilibrium
orientation, which is when its moment
p is lined up with the field E.
The expression for the potential
energy of an electric dipole in an
external electric field is simplest if we
choose the potential energy to be
zero when the angle  (Fig.22-19) is
90°.

A Dipole in an Electric Field
Potential Energy

The potential energy U of the
dipole at any other value of q can
be found by calculating the work
W done by the field on the dipole
when the dipole is rotated to that
value of q from 90°.
A Dipole in an Electric Field: Example
1-7 Shielding and Charge by Induction

Since excess charge on a conductor
is free to move, the charges will move
so that they are as far apart as
possible.

This means that excess charge on a
conductor resides on its surface, as in
the upper diagram.
1-7 Shielding and Charge by Induction
When electric charges are at rest, the electric field within a
conductor is zero.
1-7 Shielding and Charge by Induction

The electric field is always perpendicular to the surface of a
conductor – if it weren’t, the charges would move along the
surface.
1-7 Shielding and Charge by Induction

The electric field is stronger where the surface is more sharply
curved.
1-7 Shielding and Charge by Induction
A conductor can be charged by induction, if there is a way to
ground it.


This allows the like
charges to leave the
conductor;
if the conductor is
then isolated before
the rod is removed,
only the excess
charge remains.
Summary of Chapter 1
Electrons have a negative charge, and protons a positive
charge, of magnitude




Unit of charge: Coulomb, C
Charge is conserved, and quantized in units of e
Insulators do not allow electrons to move between
atoms; conductors allow conduction electrons to flow
freely
Summary of Chapter 1
The force between electric charges is along the line
connecting them
Like charges repel, opposites attract
Coulomb’s law gives the magnitude of the force:




Forces exerted by several charges add as vectors
Summary of Chapter 1
A spherical charge distribution behaves from the outside as
though the total charge were at its center
Electric field is the force per unit charge; for a point charge:



Electric fields created by several charges add as vectors
Summary of Chapter 1





Electric field lines help visualize the electric field
Field lines point in the direction of the field; start on +
charges or infinity; end on – charges or infinity; are denser
where E is greater
Parallel-plate capacitor: two oppositely charged, conducting
parallel plates
Excess charge on a conductor is on the surface
Electric field within a conductor is zero (if charges are
static)
Summary of Chapter 1



A conductor can be charged by induction
Conductors can be grounded
Electric flux through a surface:

Gauss’s law: